AE 3051, Lab #15 Investigation of the Ideal Gas State Equation By: George P. Burdell Group E3 Summer Semester 20001 Introduction An important relationship often used to relate gas properties is the ideal gas equation of state. This report describes experiments intended to investigate the validity of this state equation, which can be written as RTp , where p is the absolute pressure, T is the absolute temperature and R is the specific gas constant. In order to accomplish this, the pressure was measured for a fixed amount of gas confined in a heated, fixed volume container. The container and the gas inside were heated to a fixed temperature using an electrical resistance heater. The temperature of the gas was measured with a type-K thermocouple, and the pressure was monitored with a mercury manometer. Three gases, helium (He), nitrogen (N2) and argon (Ar), were tested. A vacuum pump was used to empty the container before it was filled with the test gas. In addition, the volume of the container was determined by filling it with water and measuring the amount with a graduated cylinder. Results Raw Data The first measurement was determination of the test cell’s volume. By filling the test cell with water, and then measuring the volume of water by pouring it into a graduated cylinder, the lab group determined its volume to be 83 ml, or equivalently, 83 cm3. Next, the lab group measured the atmospheric pressure using the digital thermometer/barometer located in the wind tunnel control room. The measured pressure was 29.66 in. Hg. Finally, the gauge pressure of the various gases as a function of temperature was measured. The results are listed in Table I for all three gases (He, N2 and Ar), with 6 measurements for each gas.2 Reduced Data The temperatures and pressures can be converted to absolute values using the following:   273 CTKT (1)       Hg.inmmHg.mmHgPaHg.inpHg.inpPapatmgage1425760101325 . (2) With the measured atmospheric pressure in the test room, Equation (2) can be simplified to:     66293386 .Hg.inpPapgage (3) The raw data listed in Table I were converted to absolute values using Equations (1) and (3). They are shown plotted in Figure 1. In order to compare the results to the ideal gas law, it is more appropriate to plot the ratio p/T , or even better, p/RT, with R given by: MRR  (4) whereRis the universal gas constant (8314 J/kmol K) and Mis the molecular weight of the gas (see Table II). The results for p/RT as a function of temperature are plotted in Figure 2, and the actual values are listed in Table III. Brief Discussion Supplement Questions Since the gas inside the container is trapped and the container volume is fixed (assuming negligible change in the size of the container due to thermal expansion during heating), the measurements for each gas sample are at the same density. According to the ideal gas state equation, we would expect the ratio p/RT to remain constant here as the gas temperature is changed. This is essentially the behavior displayed in Table III and Figure 2. Thus, we can conclude that the ratio, p/T, is proportional to density as suggested by the ideal gas law. With the current measurements, however, the proportionality constant cannot be proven to be MR.3 In order to test the remainder of the ideal gas law, one would have to determine the mass of gas trapped inside the cylinder with some independent measurement. One possibility would be to weigh the test cell twice: once with a vacuum inside and once when it is loaded with the test gas. The difference would be the weight of the test gas, which could be converted to mass using the standard value for the gravitational acceleration on Earth. We could then compare this to the mass determined from combining: 1) the p and T measurements, 2) the ideal gas law, and 3) the volume of the test cell, i.e., VRTpm  (5) However, getting an accurate value with Equation (5) would be a very difficult task. Assuming the ideal gas law is accurate, our density results from this experiment for the heaviest gas, Ar, were approximately 0.82 kg/m3. Since the volume of the test cell was measured to be 83 cm3, the mass of the Ar inside the cell was only 68 mg. It would be difficult to measure this small change in the mass of the test cell, which was approximately 2000g. In other words, an error of 0.1% in the measurement of the “empty” test cell would be ~2g. To increase the mass of gas to the level of our expected uncertainty in the test cell’s mass, the initial pressure in the cell would need to be increased from ~80 kPa (see Figure 1) to roughly 2.4 MPa or almost 350 psi. In addition, it would probably require a stronger and heavier chamber to withstand this high pressure, making the measurement even more difficult. Additional Discussion In the current experiment, we kept the density of the gas constant and changed its temperature. An alternate approach would be to use a piston-cylinder arrangement to change the volume and, therefore, the density of the gas and measure the pressure and temperature. The4 drawback to this approach is that the volume would need to change very rapidly and the temperature measurement would need to be quick enough to prevent errors due to heat exchange with the cylinder walls. In other words, if one were to rapidly compress the gas, its temperature would rise above the temperature of the cylinder walls. Very quickly, however, the gas would start losing heat to the walls and the temperature would drop back towards its original value. Tables and Figures Table I. Pressures and temperatures for 3 gases in a heated, fixed volume container. Gas T (C) pgage (in. Hg) He 22 -20.82 He 29 -20.57 He 38 -20.34 He 50 -19.96 He 65 -19.46 He 85 -18.93 N2 23 -17.88 N2 28 -17.62 N2 40 -17.20 N2 49 -16.83 N2 63 -16.23 N2 81 -15.55 Ar 22 -14.95 Ar 30 -14.48 Ar 39 -14.01 Ar 51 -13.39 Ar 63 -12.82 Ar 84 -11.84 Table II. Molecular weights of the gases measured. Gas M(kg/kmol) He 4 N2 28 Ar 405 Table III. Measurements of the ratio p/RT; also listed are the average value of the ratio for each gas. Gas T (K) p/RT (kg/m3) He 295 0.0477 He 302 0.0490 He 311 0.0488 He 323 0.0499 He 338 0.0482 He 358 0.0488 He Average 0.0487 N2 296 0.454 N2 301